A thermostatic valve of the type mentioned above has, on the one hand, the function of shortening the warm-up phase of an internal combustion engine after a cold start, i.e. to reach the operating temperature of, for example, a transmission fluid cooler for an automatic transmission. On the other hand it has the function to assure that a relatively high operating temperature is maintained and a maximally permissible temperature is not exceeded. The valve, which regulates the supply of the heated coolant to the mixing chamber, only opens when the coolant of the internal combustion engine has reached a minimum temperature on the order of 80.degree. C., for example. The valve regulating the supply of the cold or cooled coolant only opens when the operating temperature on the order of 110.degree. C. to 120.degree. C. has been reached. The thermostatic operating elements employed in the process have reached a high standard of quality today. However, it is still not possible to absolutely assure that the thermostatic valves will not fail. In this case the problem arises that both valves then remain closed, so that cooling is prevented and the transmission fluid cooler, for example, does not operate and therefore overheating in the area of the automatic transmission can occur.